We note that each of the operations considered here can be expressed in the language of the modal restriction semigroups considered in [1]}. There, algebras of partial functions in \(PT(X)\)  equipped with at least composition and “antidomain" \(A\) , where \(A(t)\)  is the identity map on \(X\)  restricted to the complement of \(\text{dom}(t)\) , \(t\in PT(X)\)  together with some or all of intersection and override (there called “preferential union") in all possible combinations, were axiomatised. Thus:
\(D(s):=A(A(s)),\ s\circ t:=D(s)t,\ s-t:=A(t)s,\ s\diamond t:=D(s)(t\sqcup s).\) 
